PHYSTAT2003, SLAC, Stanford, California, September 8-11, 2003 Statistical Challenges of Cosmic Microwave Background Analysis Benjamin D. Wandelt∗ University of Illinois at Urbana-Champaign, IL 61801, USA The Cosmic Microwave Background (CMB) is an abundant source of cosmological information. However, this information is encoded in non-trivial ways in a signal that is difficult to observe. The resulting challenges in extracting this information from CMB data sets have created a new frontier. In this talk I will discuss the challenges of CMB data analysis. I review what cosmological information is contained in the CMB data and the problem of extracting it. CMB analyses can be divided into two types: \canonical" parameter extraction which seeks to obtain the best possible estimates of cosmological parameters within a pre-defined theory space and \hypothesis testing" which seeks to test the assumption on which the canonical tests rest. Both of these activities are fundamentally important. In addition to mining the CMB for cosmological information cosmologists would like to strengthen the analysis with data from other cosmologically interesting observations as well as physical constraints. This gives an opportunity 1) to test the results from these separate probes for concordance and 2) if concordance is established to sharpen the constraints on theory space by combining the information from these separate sources. 1. OVERVIEW tered and censored/polluted measurement of this field in several `colors'. The analysis task is two-fold: in- fer the covariance structure of the field s. Infer the What is cosmic microwave background (CMB) 1 parameters Θ.” This is what could be termed \canon- statistics and what is challenging about it? It in- ical" CMB analysis. volves estimating the covariance structure of of a spa- 6 8 In this talk I will mainly describe challenges pre- tial random field with 10 {10 pixels, given only ONE sented by this canonical CMB analysis. After a brief realization of this field. The covariance matrix of these review of the scientific motivation for studying the pixels is a complicated non-linear function of the phys- CMB in section 2 I will describe the form of CMB ical parameters of interest. Of these physical param- data as well as the current status and prospects of eters there are between 10 and 20, so even finding the obtaining it in section 3. Section 4 then outlines a maximum likelihood point is hard|determining and framework for extracting cosmologically useful infor- summarizing confidence intervals around the maxi- mation from the data and section 5 illuminates some mum likelihood point is very non-trivial. Cosmolo- examples of challenges that arise when implementing gists want to do all this and have the option of building this framework. I will touch on statistical questions in exact or approximate physical constraints on rela- concerning \non-canonical" CMB analysis in section tionships between parameters. In addition, since col- 6 and then conclude in section 7. lecting cosmological data is so difficult and expensive So why are we interested in facing the statistical we want to combine all available data sets|both to challenges of CMB analysis? test them for mutual disagreement which might signal new physics, and to improve the parameter inferences. In all of this the quantification of the uncertainties in the results is extremely important|after all the 2. WHAT CAN WE LEARN FROM THE stated significance of our results will either drive or CMB? stop theoretical investigations and the design of new observational campaigns. Cosmologists are interested in studying the origins Before I get on to CMB specifics in section 2 of the physical Universe. In order to do so they have to let me give you the short version of (most) of this rely on data. For cosmologists, one of the great practi- talk for statisticians: \The CMB is an isotropic cal advantages of Einstein's relativity over Newtonian (homoschedastic) Gaussian random field s on the physics is the fact that we cannot help but look into sphere. The desired set of cosmological parameters the past. Therefore, by observing light that reaches Θ = fθi i = 1; :::; ng, are related in a non-linear way us from farther and farther away, we can study the to the spatial covariance structure Sij ≡ hsisji of the Universe directly at earlier and earlier times, at least field. Observers present us with a sampled, noisy, fil- to the extent to which the Universe is transparent to light. Since the early Universe was a hot and opaque plasma we can only see back to the time when the plasma cooled sufficiently (due to the Hubble expan- ∗NCSA Faculty Fellow 1For online material relating to this talk please refer to sion) to combine into neutral atoms and the mean http://www-conf.slac.stanford.edu/phystat2003/talks/ free time between photons collisions became of order wandelt/invited/ of the present age of the Universe. Photons that we 280 PHYSTAT2003, SLAC, Stanford, California, September 8-11, 2003 observe today which scattered for the last time in the of the CMB photons we can infer how transparent the primordial plasma are the CMB. Universe really was for the CMB photons on their way The change from plasma to gas, happened when the from last scattering to hitting our detectors. This in Universe was approximately 380,000 years old. CMB turn can tell us about the history of star formation. photons emitted at this time are therefore the most A very exciting prospect is that by studying the de- direct messengers that we can detect today of the con- tails of CMB polarization we can infer the presence or ditions present in the Universe shortly after the Big absence of gravitational waves at the time of last scat- Bang.2 They constitute a pristine snapshot of the in- tering. A detection would offer an indirect view of one fant Universe which provide us with direct cosmologi- of the elusive messengers that started their journey at cal information uncluttered by the complex non-linear an even earlier epoch, adding a nearly independent physics which led to the formation of stars and galax- constraint on the properties of the Universe at the ies. One of the main attractions of the CMB is the Planck scale. conceptual simplicity with which it can be linked to All this information is not encoded in actual fea- the global properties of the Universe and the physics tures in the CMB map of (temperature or polariza- which shaped it at or near the Planck scale. tion) anisotropies. In fact in a globally isotropic uni- As a simple example, the serendipitous discovery verse the absolute placement of individual hot and of the CMB by Penzias and Wilson in 1965 showed cold spots is devoid of useful information. Informa- that the CMB is isotropic to a very high degree. This tion can, however, be stored in the invariants of the was one of the key motivations for the development photon brightness fluctuations under the group of ro- of the inflationary paradigm [4{6]. Inflation describes tations SO(3). These are the properties of the field relative generically the emergence (from the era of quantum that only depend on the angular distance be- gravity) of a large homogeneous and isotropic Uni- tween two points of the field. For a Gaussian field, verse. Inflation also predicted the spectrum of small where 2-point statistics specify all higher order mo- metric perturbations from which later structure de- ments, this means that the angular power spectrum veloped through the gravitational instability (though coefficients of the anisotropies contain all of the infor- it was not the only mechanism to do so). The corre- mation. sponding anisotropies in the CMB were first convinc- The challenge for theoreticians was then to develop ingly detected by the DMR instrument on the COBE a detailed theory of the angular power spectrum C`, satellite in 1992. The rejection of alternative mecha- as a function of angular wavenumber `, given the cos- nisms for the generation of the primordial spectrum mological parameters Θ. While conceptually simple, of metric perturbations in favor of inflation was a ma- it required a decade-long intellectual effort to model jor advance driven by the measurement of the large the relevant physical processes at the required level of angle CMB anisotropy. These developments have led precision. As a result, there now exist several Boltz- to inflation becoming part of the current cosmological mann codes (e.g. CMBFAST [2] or CAMB [3]), which standard model. A very robust prediction of generic numerically compute C`(Θ) to 1% precision or bet- model implementations of inflation is the Gaussianity ter. The power spectra C`(Θ) are sensitive functions and homogeneity of the resulting perturbations. of certain combinations of the parameters and weak functions of others (degeneracies). These weakly con- Quantitatively, the properties of our Universe are strained parameter combinations are referred to as de- ∼ − cosmological parameters encoded in a set of n 10 20 generacies. Within the context of the standard cos- Θ where n depends on the level of detail of the mod- mological model, the theory of this dependence is well- elling or, commonly, on the specification of theoretical understood. priors which fix some of these parameters to \reason- It is clear from the preceding discussion that the able" values. These parameters specify the geometry CMB is an extremely valuable source of what amounts and average energy density of the Universe, as well as to \cosmological gold": information about the physics the relative amounts of energy density contributed by and the global properties of the early Universe. So the ingredients of the primordial soup (dark matter, what are the observational prospects? ordinary baryons, neutrinos and dark energy and pho- tons).